The efficient computation of bounds for functionals of finite element solutions in large strain elasticity

A new residual-type flux-free error estimator is presented. It computes upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux-equilibration and resulting in a simple implementation that uses standard resources available in finite element codes. This is especially interesting for existing codes and 3D applications where the implementation of this technique is as simple as in 2D. Recall that on the contrary, the complexity of the flux-equilibration techniques increases drastically in the 3D case. Bounds for the energy norm of the error are used to produce upper and lower bounds of linear functional outputs, representing quantities of engineering interest. This new flux-free error estimator improves the effectivity of previous approaches (better accuracy in every test) and it can be used in the mechanical case for linear elements. The proposed approach demonstrates its efficiency in numerical tests producing sharp bounds of the reference error both for the energy and the quantities of interest.

show abstract

“…a finer mesh). Bounds for the exact solution of the boundary value problem as presented in [4][5][6] are not addressed here.…”

Section: Introductionmentioning

confidence: 99%

Subdomain-based flux-free a posteriori error estimators

Mariné

Dı́ez

Huerta

2006

Computer Methods in Applied Mechanics and Engineering

100

171

View full text Add to dashboard Cite

show abstract

“…Analogously to finite elements an adjoint problem is introduced to obtain a proper error representation, see (5). The adjoint problem consists in finding ψ ∈ V such that…”

Section: Estimation Of Outputs Of Interestmentioning

confidence: 99%

“…1783 and a much finer discretization). Bounds for the exact solution of the boundary value problem as presented in [5][6][7][8][9] are not addressed here.The need for obtaining reliable upper and lower bounds of the error for quantities of interest has motivated the use of implicit residual error estimators, which are currently the only type of estimators ensuring bounds for the error. Classical residual-type estimators, which provide upper bounds of the error, require flux-equilibration procedures (hybrid-flux techniques) to properly set boundary conditions for local problems [2, 10].…”

mentioning

confidence: 99%

Bounds for quantities of interest and adaptivity in the element‐free Galerkin method

Vidal

Mariné

Dı́ez

et al. 2008

Numerical Meth Engineering

View full text Add to dashboard Cite

1 of estimators ensuring bounds for the error. Classical residual type estimators, which provide upper bounds of the error, require flux-equilibration procedures (hybrid-flux techniques) to properly set boundary conditions for local problems [10, 2]. Flux-equilibration requires a domain decomposition, which is natural in finite elements but not in mesh-free methods. Moreover, it is performed by a complex algorithm, strongly dependent on the finite element type and requiring a data structure that is not natural in a standard finite element code.The idea of using flux-free estimators, based on the partition-of-the-unity concept and using local subdomains different than elements, has been already proposed in [11,12,13,14] for finite elements. The main advantage of the flux-free approach is the simplicity in the implementation. Obviously, this is especially important in the 3D case.From the mesh-free point of view, another advantage is the fact that the local subdomains where the error equation is solved are the support of the functions characterizing the partition of unity. This is a concept that also exists in mesh-free methods and thus the extension is possible. Moreover, boundary conditions of the local problems are natural and the usual data structure of a code is directly employed.In the last few years, some research has been devoted to develop error estimation procedures for mesh-free methods. To the authors knowledge implicit residual based estimators have not been proposed for mesh-free methods. However, in [19] a strategy to assess the energy norm of the error for the Generalized Finite Element Method (GFEM) is presented and the possible generalizations to mesh-free methods are also devised. Implicit residual error estimators are now standard in finite elements because they are mathematically sound, more accurate and allow computing upper and lower bounds for energy norms as well as for functional outputs. In this paper the implicit residual-type flux-free error estimator proposed in [14], which is comparable in effectivity to the standard hybrid-flux estimators, is extended to the Element Free Galerkin Method. In the present work, the difficulties found when extending the implicit residual methods to mesh-free are related with the fact that the refined discretization do not induce nested functional spaces. This is specially important for the error representation. Moreover, the bounds for the specific quantity of interest must be in this case carefully computed. In reference [19] these difficulties are not encountered because the error estimator is performed in terms of the energy norm and in the GFEM the refined subspaces are nested.Mesh-free methods are especially well suited for adaptivity. Enriching the mesh-free discretization is straightforward because the new particles are added without any connectivity restriction. Here, a refinement scheme based on inserting particles following a quadtree structure in a background rectangular grid is proposed. The refinement criterion uses the information provided...

show abstract

“…For global error estimators in finite elasticity, we refer to Brink and Stein (1998), whereas goal-oriented a posteriori error estimators were developed by Rüter et al (2001Rüter et al ( , 2004b, Bonet, Huerta and Peraire (2002), and Rüter, Ohnimus and Stein (2004a).…”

Section: Introductionmentioning

confidence: 99%

Finite Element Methods for Elasticity with Error‐Controlled Discretization and Model Adaptivity

Stein¹,

Rüter²

2004

Encyclopedia of Computational Mechanics

View full text Add to dashboard Cite

The essential topics of the finite element method for linear and finite elastic deformations of solids are presented in this chapter from both the mechanical and mathematical point of view. As a starting point, the nonlinear, and linearized theory of elasticity are derived in a rigorous way, followed by the classical variational principles of elasticity, which are the basis for the finite element method in its various forms. More precisely, the discrete variational approach of the (one‐field) Dirichlet minimization principle of the total potential energy is presented, followed by concise representations of the (two‐field) Hellinger–Reissner stationary dual‐mixed principle and the (three‐field) Hu–Washizu stationary mixed principle, including the main features of the associated finite element methods. The main objective of this chapter is the systematic treatment of error estimation procedures and adaptivity for the linearized and finite elasticity problem covering both global and goal‐oriented a posteriori error estimators. The three basic classes, that is, residual‐, hierarchical‐, and averaging‐type error estimators are presented and applied to a fracture mechanics problem as an example. A further challenging topic is the combination of error‐controlled adaptive finite element solutions with hierarchical model and dimension adaptivity of the underlying mathematical model, especially for thin‐walled structures where model expansion is necessary in subdomains with boundary layers and other disturbances.

show abstract

The efficient computation of bounds for functionals of finite element solutions in large strain elasticity

Cited by 10 publications

References 21 publications

Subdomain-based flux-free a posteriori error estimators

Subdomain-based flux-free a posteriori error estimators

Bounds for quantities of interest and adaptivity in the element‐free Galerkin method

Finite Element Methods for Elasticity with Error‐Controlled Discretization and Model Adaptivity

Contact Info

Product

Resources

About